Contributions to the verification and control of timed and probabilistic models Nathalie Bertrand Inria Rennes Habilitation defense - November 16th 2015 November 16th 2015 – Habilitation Defense
Formal verification of software systems Software systems are everywhere. Bugs are everywhere. Formal verification should be everywhere! static analysis analysis of the source code of a program in a static manner, i.e. without executing it theorem proving automated proofs of mathematical statements through logical reasoning using deduction rules model based testing generation of a set of testing scenarios, given a model of the system model checking certification that a mathematical representation of the system satisfies a model of its specification Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 2/ 30
Formal verification of software systems Software systems are everywhere. Bugs are everywhere. Formal verification should be everywhere! static analysis analysis of the source code of a program in a static manner, i.e. without executing it theorem proving automated proofs of mathematical statements through logical reasoning using deduction rules model based testing generation of a set of testing scenarios, given a model of the system model checking certification that a mathematical representation of the system satisfies a model of its specification Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 2/ 30
Principles of model checking satisfy Does ? system specification Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 3/ 30
Principles of model checking satisfy Does ? system specification ϕ model formula Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 3/ 30
Principles of model checking satisfy Does ? system specification | ϕ = ? model checker model formula Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 3/ 30
Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 4/ 30
Rich models for complex systems 12 11 1 10 2 delays, timeouts 9 3 8 4 real-time systems 7 5 6 timing constraints Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 5/ 30
Rich models for complex systems 12 11 1 10 2 delays, timeouts randomized algorithms 9 3 8 4 real-time systems unpredictable behaviours 7 5 6 timing constraints probabilities Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 5/ 30
Rich models for complex systems 12 11 1 10 2 delays, timeouts randomized algorithms 9 3 8 4 real-time systems unpredictable behaviours 7 5 6 timing constraints probabilities large systems security concerns partial observation Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 5/ 30
Rich models for complex systems 12 11 1 10 2 delays, timeouts randomized algorithms 9 3 8 4 real-time systems unpredictable behaviours 7 5 6 timing constraints probabilities large systems unknown value security concerns generic systems parameters partial observation Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 5/ 30
Contributions in a nutshell model-based testing model checking monitoring issues 12 11 1 controller synthesis 10 2 9 3 8 4 7 5 6 decidability complexity algorithms Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 6/ 30
Outline ❶ timed automata 12 11 1 10 2 9 3 8 4 7 5 6 Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 7/ 30
Outline ❶ timed automata 12 11 1 ❷ stochastic timed automata 10 2 9 3 8 4 7 5 6 Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 7/ 30
Outline ❶ timed automata 12 11 1 ❷ stochastic timed automata 10 2 9 3 8 4 7 5 6 ❸ partially observable probabilistic systems Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 7/ 30
Outline ❶ timed automata 12 11 1 ❷ stochastic timed automata 10 2 9 3 8 4 7 5 6 ❸ partially observable probabilistic systems ❹ parameterized probabilistic networks Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 7/ 30
Outline ❶ timed automata 12 11 1 10 2 9 3 8 4 7 5 6 Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 8/ 30
12 11 1 10 2 9 3 Determinizing timed automata 8 4 7 5 6 0 < x < 1 , a ℓ 1 0 a , < 1 < x < x < 1 , 0 b ℓ 0 ℓ 3 0 < x < 1 , a x b , : = 0 = 0 x ℓ 2 ( a ,. 5 )( b ,. 5 ) read on two paths Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 9/ 30
12 11 1 10 2 9 3 Determinizing timed automata 8 4 7 5 6 0 < x < 1 , a ℓ 1 0 < y < 1 , a , z := 0 0 a , < 1 < x < x < 1 , 0 b 0 < y < 1 , a 0 ≤ z < y < 1 , b ℓ 0 ℓ 3 ℓ 0 ℓ 1 ℓ 2 0 < x z := 0 < 1 , a x b , : = 0 = 0 x ℓ 2 ( a ,. 5 )( b ,. 5 ) read on two paths Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 9/ 30
12 11 1 10 2 9 3 Determinizing timed automata 8 4 7 5 6 0 < x < 1 , a ℓ 1 0 < y < 1 , a , z := 0 0 a , < 1 < x < x < 1 , 0 b 0 < y < 1 , a 0 ≤ z < y < 1 , b ℓ 0 ℓ 3 ℓ 0 ℓ 1 ℓ 2 0 < x z := 0 < 1 , a x b , : = 0 = 0 x ℓ 2 ( a ,. 5 )( b ,. 5 ) read on two paths Motivations for determinization simpler model, easy complementation, offline monitor synthesis Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 9/ 30
12 11 1 10 2 9 3 Determinizing timed automata 8 4 7 5 6 0 < x < 1 , a ℓ 1 0 < y < 1 , a , z := 0 0 a , < 1 < x < x < 1 , 0 b 0 < y < 1 , a 0 ≤ z < y < 1 , b ℓ 0 ℓ 3 ℓ 0 ℓ 1 ℓ 2 0 < x z := 0 < 1 , a x b , : = 0 = 0 x ℓ 2 ( a ,. 5 )( b ,. 5 ) read on two paths Motivations for determinization simpler model, easy complementation, offline monitor synthesis Hard problem for timed automata ◮ determinization unfeasible in general ◮ determinizability undecidable [AD94] Alur and Dill, A theory of timed automata . TCS, 1994. [Tri06] Tripakis, Folk theorems on the determinization and minimization of timed automata , IPL, 2006. [Fin06] Finkel, Undecidable problems about timed automata , Formats’06. Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 9/ 30
12 11 1 10 2 9 3 Game-based over-approximation algorithm 8 4 7 5 6 [FoSSaCS’11, FMSD’15] ℓ 0 , x − y = 0 , ⊤ { 0 } w. J´ eron, Krichen, Stainer ( 0 , 1 ) , a Am´ elie Stainer’s PhD thesis ∅ { y } { y } { 0 } , a ∅ ◮ exact determinization or ℓ 0 , x − y = 0 , ⊤ ℓ 0 , 0 < x − y < 1 , ⊤ ℓ 1 , x − y = 0 , ⊤ (0,1) ℓ 1 , 0 < x − y < 1 , ⊤ { 0 } ℓ 2 , − 1 < x − y < 0 , ⊤ ℓ 2 , x − y = 0 , ⊤ over-approximation { 0 } , b ( 0 , 1 ) , a ∅ { y } ( 0 , 1 ) , b ◮ subsumes exact ( 0 , 1 ) ∅ { y } , a ∅ ℓ 3 , x − y = 0 , ⊤ { y } determinization procedure { 0 } ℓ 3 , x − y = 0 , ⊥ ( 0 , 1 ) , a { y } { y } ∅ ℓ 3 , x − y = 0 , ⊤ { 0 } w. Baier, Bouyer and ℓ 0 , 0 < x − y < 1 , ⊥ ℓ 0 , 0 < x − y < 1 , ⊥ ∅ ℓ 1 , 0 < x − y < 1 , ⊥ (0,1) ℓ 1 , 0 < x − y < 1 , ⊥ { 0 } { y } ℓ 2 , − 1 < x − y < 0 , ⊥ ℓ 2 , x − y = 0 , ⊥ Brihaye [ICALP’09] ∅ ( 0 , 1 ) , b ( 0 , 1 ) , a ℓ 3 , − 1 < x − y < 0 , ⊤ { y } { 0 } , b (0,1) { 0 } , a ◮ no complexity overhead ℓ 3 , − 1 < x − y < 0 , ⊥ ∅ ( 0 ∅ , ( 0 , 1 ) , b 1 ) , b { y } ◮ application to offline test ℓ 3 , x − y = 0 , ⊥ { 0 } { y } { y } { y } generation w. J´ eron, Krichen and Stainer ∅ ∅ ∅ [TACAS’11, LMCS’12] ℓ 3 , 0 < x − y < − 1 , ⊥ ( 0 , 1 ) Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 10/ 30
Outline 12 11 1 ❷ stochastic timed automata 10 2 9 3 8 4 7 5 6 Verification and control of quantitative models – Nathalie Bertrand November 16th 2015 – Habilitation Defense – 11/ 30
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